Inequality in the sense of distributions
1
$begingroup$
What does
$$
fleq g
$$
mean for two Schwartz distributions/generalized functions $f $ and $g$? I assume it means that the inequality holds when tested with positive functions. The next question would then be: if such an inequality holds, is it always true that $g-f$ is a positive "proper" function ( $in L^1_{loc}$) or measure ?
analysis pde schwartz-space
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
$endgroup$
add a comment |
1
$begingroup$
What does
$$
fleq g
$$
mean for two Schwartz distributions/generalized functions $f $ and $g$? I assume it means that the inequality holds when tested with positive functions. The next question would then be: if such an inequality holds, is it always true that $g-f$ is a positive "proper" function ( $in L^1_{loc}$) or measure ?
analysis pde schwartz-space
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
$endgroup$
2
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
2
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36
add a comment |
1
1
1
$begingroup$
What does
$$
fleq g
$$
mean for two Schwartz distributions/generalized functions $f $ and $g$? I assume it means that the inequality holds when tested with positive functions. The next question would then be: if such an inequality holds, is it always true that $g-f$ is a positive "proper" function ( $in L^1_{loc}$) or measure ?
analysis pde schwartz-space
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
$endgroup$
What does
$$
fleq g
$$
mean for two Schwartz distributions/generalized functions $f $ and $g$? I assume it means that the inequality holds when tested with positive functions. The next question would then be: if such an inequality holds, is it always true that $g-f$ is a positive "proper" function ( $in L^1_{loc}$) or measure ?
analysis pde schwartz-space
analysis pde schwartz-space
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
asked Dec 5 '18 at 12:00
BananachBananach
3,86611429
3,86611429
2
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
2
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36
add a comment |
2
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
2
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36
2
2
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
2
2
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36
add a comment |
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2
$begingroup$
If $fge0$ then it can be represented as a measure. Reference: Hormander 'The Analysis of Partial Differential Operators - Vol 1' Thm 2.1.7
$endgroup$
– daw
Dec 5 '18 at 12:31
2
$begingroup$
For $u in C^0_c(K)$ take a sequence of test functions $phi_n$ decreasing to $u$. Then $f(phi_n)$ is decreasing and $ge 0$ thus it converges, and choosing $phi_0 = 2|u|_infty chi_K$ ($chi_K$ a $C^infty_c$ approximation of $1_K$) we have $|f(phi_n)|le |f(chi_K)|2 |u|_infty$ thus $f$ is a measure @daw
$endgroup$
– reuns
Dec 5 '18 at 13:10
$begingroup$
@reuns if you turn this into an answer, I will accept it
$endgroup$
– Bananach
Dec 6 '18 at 8:36